Algebra Examples

$f (x) = 3 + 2 x^{2} - 4 x^{3}$

Step 1

Find the point at $x = - 1$ .

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Step 1.1

Replace the variable $x$ with $- 1$ in the expression.

$f (- 1) = 3 + 2 {(- 1)}^{2} - 4 {(- 1)}^{3}$

Step 1.2

Simplify the result.

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Step 1.2.1

Simplify each term.

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Step 1.2.1.1

Raise $- 1$ to the power of $2$ .

$f (- 1) = 3 + 2 \cdot 1 - 4 {(- 1)}^{3}$

Step 1.2.1.2

Multiply $2$ by $1$ .

$f (- 1) = 3 + 2 - 4 {(- 1)}^{3}$

Step 1.2.1.3

Raise $- 1$ to the power of $3$ .

$f (- 1) = 3 + 2 - 4 \cdot - 1$

Step 1.2.1.4

Multiply $- 4$ by $- 1$ .

$f (- 1) = 3 + 2 + 4$

Step 1.2.2

Simplify by adding numbers.

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Step 1.2.2.1

Add $3$ and $2$ .

$f (- 1) = 5 + 4$

Step 1.2.2.2

Add $5$ and $4$ .

$f (- 1) = 9$

Step 1.2.3

The final answer is $9$ .

$9$

Step 1.3

Convert $9$ to decimal.

$y = 9$

Step 2

Find the point at $x = 0$ .

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Step 2.1

Replace the variable $x$ with $0$ in the expression.

$f (0) = 3 + 2 {(0)}^{2} - 4 {(0)}^{3}$

Step 2.2

Simplify the result.

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Step 2.2.1

Simplify each term.

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Step 2.2.1.1

Raising $0$ to any positive power yields $0$ .

$f (0) = 3 + 2 \cdot 0 - 4 {(0)}^{3}$

Step 2.2.1.2

Multiply $2$ by $0$ .

$f (0) = 3 + 0 - 4 {(0)}^{3}$

Step 2.2.1.3

Raising $0$ to any positive power yields $0$ .

$f (0) = 3 + 0 - 4 \cdot 0$

Step 2.2.1.4

Multiply $- 4$ by $0$ .

$f (0) = 3 + 0 + 0$

Step 2.2.2

Simplify by adding numbers.

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Step 2.2.2.1

Add $3$ and $0$ .

$f (0) = 3 + 0$

Step 2.2.2.2

Add $3$ and $0$ .

$f (0) = 3$

Step 2.2.3

The final answer is $3$ .

$3$

Step 2.3

Convert $3$ to decimal.

$y = 3$

Step 3

Find the point at $x = 1$ .

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Step 3.1

Replace the variable $x$ with $1$ in the expression.

$f (1) = 3 + 2 {(1)}^{2} - 4 {(1)}^{3}$

Step 3.2

Simplify the result.

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Step 3.2.1

Simplify each term.

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Step 3.2.1.1

One to any power is one.

$f (1) = 3 + 2 \cdot 1 - 4 {(1)}^{3}$

Step 3.2.1.2

Multiply $2$ by $1$ .

$f (1) = 3 + 2 - 4 {(1)}^{3}$

Step 3.2.1.3

One to any power is one.

$f (1) = 3 + 2 - 4 \cdot 1$

Step 3.2.1.4

Multiply $- 4$ by $1$ .

$f (1) = 3 + 2 - 4$

Step 3.2.2

Simplify by adding and subtracting.

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Step 3.2.2.1

Add $3$ and $2$ .

$f (1) = 5 - 4$

Step 3.2.2.2

Subtract $4$ from $5$ .

$f (1) = 1$

Step 3.2.3

The final answer is $1$ .

$1$

Step 3.3

Convert $1$ to decimal.

$y = 1$

Step 4

Find the point at $x = 2$ .

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Step 4.1

Replace the variable $x$ with $2$ in the expression.

$f (2) = 3 + 2 {(2)}^{2} - 4 {(2)}^{3}$

Step 4.2

Simplify the result.

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Step 4.2.1

Simplify each term.

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Step 4.2.1.1

Multiply $2$ by ${(2)}^{2}$ by adding the exponents.

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Step 4.2.1.1.1

Multiply $2$ by ${(2)}^{2}$ .

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Step 4.2.1.1.1.1

Raise $2$ to the power of $1$ .

$f (2) = 3 + 2 {(2)}^{2} - 4 {(2)}^{3}$

Step 4.2.1.1.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f (2) = 3 + 2^{1 + 2} - 4 {(2)}^{3}$

Step 4.2.1.1.2

Add $1$ and $2$ .

$f (2) = 3 + 2^{3} - 4 {(2)}^{3}$

Step 4.2.1.2

Raise $2$ to the power of $3$ .

$f (2) = 3 + 8 - 4 {(2)}^{3}$

Step 4.2.1.3

Raise $2$ to the power of $3$ .

$f (2) = 3 + 8 - 4 \cdot 8$

Step 4.2.1.4

Multiply $- 4$ by $8$ .

$f (2) = 3 + 8 - 32$

Step 4.2.2

Simplify by adding and subtracting.

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Step 4.2.2.1

Add $3$ and $8$ .

$f (2) = 11 - 32$

Step 4.2.2.2

Subtract $32$ from $11$ .

$f (2) = - 21$

Step 4.2.3

The final answer is $- 21$ .

$- 21$

Step 4.3

Convert $- 21$ to decimal.

$y = - 21$

Step 5

Find the point at $x = 3$ .

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Step 5.1

Replace the variable $x$ with $3$ in the expression.

$f (3) = 3 + 2 {(3)}^{2} - 4 {(3)}^{3}$

Step 5.2

Simplify the result.

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Step 5.2.1

Simplify each term.

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Step 5.2.1.1

Raise $3$ to the power of $2$ .

$f (3) = 3 + 2 \cdot 9 - 4 {(3)}^{3}$

Step 5.2.1.2

Multiply $2$ by $9$ .

$f (3) = 3 + 18 - 4 {(3)}^{3}$

Step 5.2.1.3

Raise $3$ to the power of $3$ .

$f (3) = 3 + 18 - 4 \cdot 27$

Step 5.2.1.4

Multiply $- 4$ by $27$ .

$f (3) = 3 + 18 - 108$

Step 5.2.2

Simplify by adding and subtracting.

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Step 5.2.2.1

Add $3$ and $18$ .

$f (3) = 21 - 108$

Step 5.2.2.2

Subtract $108$ from $21$ .

$f (3) = - 87$

Step 5.2.3

The final answer is $- 87$ .

$- 87$

Step 5.3

Convert $- 87$ to decimal.

$y = - 87$

Step 6

The cubic function can be graphed using the function behavior and the points.

$\begin{array}{c} x & y \\ - 1 & 9 \\ 0 & 3 \\ 1 & 1 \\ 2 & - 21 \\ 3 & - 87 \end{array}$

Step 7

The cubic function can be graphed using the function behavior and the selected points.

Rises to the left and falls to the right

$\begin{array}{c} x & y \\ - 1 & 9 \\ 0 & 3 \\ 1 & 1 \\ 2 & - 21 \\ 3 & - 87 \end{array}$

Step 8